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Vladimir Tarkaev

Publications and source records attributed to Vladimir Tarkaev.

7 recordsLinked to original sources
arXiv2024

A prime decomposition theorem for string links in a thickened surface

We prove a prime decomposition theorem for string links in a thickened surface. Namely, we prove that any non-braid string link $\ell \subset Σ\times I$, where $Σ$ is a compact orientable (not necessarily closed) surface other than $S^2$, can be written in the form $\ell =\ell_1 \# \ldots \# \ell_m$, where $\ell_j,j=1,\ldots,m,$ is prime string link defined up to braid equivalence, and the decomposition is unique up to possibly permuting the order of factors in its right-hand side.

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arXiv2024

An analog of the {K}auffman bracket polynomial for knots in the non-orientable thickening of a non-orientable surface

We study pseudo-classical knots in the non-orientable thickening of a non-orientable surface, specifically knots that are orientation-preserving paths in a non-orientable $3$-manifold of the form (non-orientable surface) $\times$ $[0, 1]$. For these knots, we propose an analog of the Kauffman bracket polynomial. The construction of this polynomial closely mirrors the classical version, with key differences in the definitions of the sign of a crossing and the positive/negative smoothing of a crossing. We prove that this polynomial is an isotopy invariant of pseudo-classical knots and demonstrate that it is independent of the classical Kauffman bracket polynomial for knots in the thickened orientable surface, which is the orientable double cover of the non-orientable surface under consideration.

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arXiv2024

An analogue of Turaev comultiplication for knots in non-orientable thickening of a non-orientable surface

This paper concerns pseudo-classical knots in the non-orientable manifold $\hatΣ =Σ\times [0,1]$, where $Σ$ is a non-orientable surface and a knot $K \subset \hatΣ$ is called pseudo-classical if $K$ is orientation-preserving path in $\hatΣ$. For this kind of knot we introduce an invariant $Δ$ that is an analogue of Turaev comultiplication for knots in a thickened orientable surface. As its classical prototype, $Δ$ takes value in a polynomial algebra generated by homotopy classes of non-contractible loops on $Σ$, however, as a ground ring we use some subring of $\mathbb{C}$ instead of $\mathbb{Z}$. Then we define a few homotopy, homology and polynomial invariants, which are consequences of $Δ$, including an analogue of the affine index polynomial.

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arXiv2020

Homological Casson type invariant of knotoids

We consider an analogue of well-known Casson knot invariant for knotoids. We start with a direct analogue of the classical construction which gives two different integer-valued knotoid invariants and then focus on its homology extension. Value of the extension is a formal sum of subgroups of the first homology group $H_1(Σ)$ where $Σ$ is an oriented surface with (maybe) non-empty boundary in which knotoid diagrams lie. To make the extension informative for spherical knotoids it is sufficient to transform an initial knotoid diagram in $S^2$ into a knotoid diagram in the annulus by removing small disks around its endpoints. As an application of the invariants we prove two theorems: a sharp lower bound of the crossing number of a knotoid (the estimate differs from its prototype for classical knots proved by M.Polyak and O.Viro in 2001) and a sufficient condition for a knotoid in $S^2$ to be a proper knotoid (or pure knotoid with respect to Turaev's terminology). Finally we give a table containing values of our invariants computed for all spherical prime proper knotoids having diagrams with at most $5$ crossings.

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arXiv2020

A relation between the crossing number and the height of a knotoid

Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by V.~Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such an arcs disjoint from crossings. In the paper we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.

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arXiv2015

A census of tetrahedral hyperbolic manifolds

We call a cusped hyperbolic 3-manifold tetrahedral if it can be decomposed into regular ideal tetrahedra. Following an earlier publication by three of the authors, we give a census of all tetrahedral manifolds and all of their combinatorial tetrahedral tessellations with at most 25 (orientable case) and 21 (non-orientable case) tetrahedra. Our isometry classification uses certified canonical cell decompositions (based on work by Dunfield, Hoffman, Licata) and isomorphism signatures (an improvement of dehydration sequences by Burton). The tetrahedral census comes in Regina as well as SnapPy format, and we illustrate its features.

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